Right now, Washington looks and feels stuck in No Way Out syndrome. It has nothing to do with that Cold War celluloid thriller and everything to do with ideology and inertia in the debt-ceiling debate. By all outward appearances, the dominant caucuses in Congress can be named “I Won’t” and “I Can’t.”

But is there room for an 11th-hour compromise? Is the glue available, possibly even visible, to patch together a deal?

Possibly.

First, of course, the existing plans have to die their inevitable legislative deaths. House Speaker John Boehner’s $900 billion debt-ceiling increase, now redrafted with $917 billion in cuts certified by the Congressional Budget Office, will pass the House on Thursday. It will then die when the Democratically controlled Senate tables it. Senate Majority Leader Harry Reid’s $2.7 trillion debt-ceiling increase will not surmount a GOP filibuster and therefore will die as well.

After swift burial, Congress will have four days to work out a deal to avert the default that Treasury says will begin on August 2. The key sticking point and the must-have for President Obama and Reid is an increase in debt authority of at least $2.7 trillion—enough to last until early 2013. When Reid sent a letter to Boehner on Wednesday signed by all 51 Senate Democrats and the two independents promising to kill his bill, the only reason cited was that it lacked a debt-limit increase past the 2012 election. This is Obama and Reid’s irreducible minimum. Democrats cannot and will not back away from it—not for economic or political reasons which, for all Democrats, are tightly intertwined.

Republicans must reconcile themselves to this fact. The question, therefore, is this: can both sides strike a deal and avert a default? It won’t be easy. To do so, Obama and Reid will have to give House Republicans additional concessions—compared to where things stand now—to draft a bill that can pass in both chambers. Boehner might have to give a thing or two as well.

Based on National Journal's reporting, a smidgen of tea leaf reading, and a careful look at the underlying numbers and policy, here are five areas of possible compromise.

1. Balanced Budget Amendment: House Republicans desperately want a vote in both chambers on a balanced-budget amendment to the Constitution. It’s required in the Boehner bill but Reid’s bill doesn’t address it. Adding a balanced-budget amendment vote in the Senate (where Democrats would offer an alternative version) could mollify some House Republicans.

2. Tax increases: The Boehner bill establishes a 12-member congressional "Super Committee" to draft $1.8 trillion in deficit reduction this year and gives the committee’s recommendations an expedited path to the House and Senate floor with no opportunity for amendments. For some mysterious reason, Boehner’s bill does not tell the committee it can’t raise taxes—even though Reid has offered $2.7 trillion in deficit reduction without tax increases. If the committee mandate was to pursue tax reform in the context of deficit reduction (a stated goal of both parties) but do so without a net tax increase, Boehner might be able to sell House Republicans on a longer debt-ceiling increase.

3. Defense spending: The Reid bill seeks much less in security spending (defense and veterans) than House Republicans and Obama. The numbers appear artificially low, seeking defense cuts that Reid might not get through the Senate when considered in isolation. The apparent artificiality of the numbers suggests they might be fodder for compromise. Reid, for example, seeks $31 billion less for security than Obama in fiscal year 2012 and $49 billion less than Obama in 2013 ($29 billion in 2012 and $47 billion less in 2013 when compared to the budget authored by Rep. Paul Ryan, R-Wis.). If Reid agreed to higher defense numbers, that might prove attractive to House Republicans—but he would have to give a lot, thus jeopardizing non-defense discretionary spending. At this late hour, these trade-offs will almost have to be zero-sum maneuvers.